Hall's conjecture says that for every $\epsilon>0$ there is a $C_\epsilon$ such that if $x$ and $y$ are integers with $x^3-y^2\ne0$, then $$ |x^3 - y^2| \ge C_\epsilon \max\{|x^3|,|y^2|\}^{1/6-\epsilon}, $$ so this would imply that the separation indeed gets quite large as $x$ and $y$ increase. The polynomial version, i.e., when $x$ and $y$ are in $\mathbb{C}[T]$, was proven by Davenport with $\epsilon=0$, i.e., if $x^3 - y^2 \ne 0$ then $$ \deg(x^3-y^2) \ge \frac16 \max\{\deg(x^3),\deg(y^2)\} + C $$ for an absolute constant $C$ (in fact, we can use $C=1$). More generally in the polynomial (or function field) case, one can easily prove lower bounds for $\deg(x^n-y^m)$, with analogous conjectures over $\mathbb{Z}$ (or over number fields).
Addendum: As Ryan D'Mello points out, the above doesn't really answer the question, which asks about gaps between $x$ values of solutions to $0<|x^3-y^2|<\sqrt{x}$. However, I think that in order to get a reasonable answer, one will need to assume something like Hall's conjecture. Alternatively, for a given (small) $\epsilon>0$, one might hope to unconditionally prove a gap estimate for the set $$ \bigl\{ x\in\mathbb{Z} : \text{there exists $y\in\mathbb{Z}$ with $|x|^{1/2-\epsilon}<|x^3-y^2|<|x|^{1/2+\epsilon}$} \bigr\}, $$ where the size of the gaps depends on $\epsilon$.